Granular Matter (2009) 11:139–156 DOI 10.1007/s10035-009-0137-3 Granular solid hydrodynamics Yimin Jiang · Mario Liu Received: 21 October 2008 / Published online: 21 April 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com Abstract A complete continuum mechanical theory for 5 Elastic and plastic strain .................... 146 granular media, including explicit expressions for the energy 6 The granular free energy .................... 146 current and the entropy production, is derived and explained. 6.1 The elastic energy ..................... 147 6.2 Density dependence of B ................. 148 Its underlying notion is: granular media are elastic when at 6.3 Higher-order strain terms ................. 148 rest, but turn transiently elastic when the grains are agitated— 6.4 Pressure contribution from agitated grains ........ 149 such as by tapping or shearing. The theory includes the true 6.5 The Edwards entropy ................... 150 temperature as a variable, and employs in addition a gran- 7 Granular hydrodynamic theory ................ 150 ular temperature to quantify the extent of agitation. A free 7.1 Derivation ......................... 150 energy expression is provided that contains the full jamming 7.2 Results ........................... 152 8 The hypoplastic regime .................... 152 phase diagram, in the space spanned by pressure, shear stress, 9 Conclusion ........................... 153 density and granular temperature. We refer to the theory as GSH, for granular solid hydrodynamics. In the static limit, it reduces to granular elasticity, shown previously to yield realistic static stress distributions. For steady-state defor- 1 Introduction mations, it is equivalent to hypoplasticity, a state-of-the-art engineering model. Widespread interests in granular media were aroused among physicists a decade ago, stimulated in large part by review articles revealing the intriguing fact that something as Contents familiar as sand is still rather poorly understood [1–4]. The resultant collective efforts since have greatly enhanced our 1 Introduction .......................... 139 understanding, though the majority of theoretic consider- 2 Sand: a transiently elastic medium .............. 142 ations are focused either on the limit of highly excited 3 Granular equilibria and jamming ............... 143 3.1 Liquid equilibrium .................... 143 gaseous state [5–10], or that of the fluid-like flow [11–13]. 3.2 Solid equilibrium ..................... 143 Except in some noteworthy and insightful simulations 3.3 Granular equilibria .................... 143 [14–16], the quasi-static, elasto-plastic motion of dense gran- 4 Granular temperature Tg .................... 144 ular media—of technical relevance and hence a reign of 4.1 The equilibrium condition for T ............. 144 g engineers—received less attention among physicists. This 4.2 The equation of motion for sg ............... 144 4.3 Two fluctuation–dissipation theorems ........... 145 choice is due (at least in part) to the confusing state of engi- neering theories, where innumerable continuum mechanical Y. Jiang · M. Liu (B) models compete, employing strikingly different expressions. Theoretische Physik, Universität Tübingen, Although the better ones achieve considerable realism when 72076 Tübingen, Germany confined to the effects they were constructed for, these differ- e-mail: [email protected] ential equations are more a rendition of complex empirical Y. Jiang data, less a reflection of the underlying physics. In a forth- Central South University, 410083 Changsha, China coming book on soil mechanics by Gudehus, phrases such 123 140 Y. Jiang, M. Liu as morass of equations and jungle of data were used as met- itly, its full functional dependence. Hydrodynamic theories aphors. may well be derived without the second step, and the result Most engineering theories are elasto-plastic [17–19], is referred to as the structure of the theory. The form of all though there are also hypoplastic ones, which manage to fluxes, including especially the stress σij, are in this case retain the realism while being simpler and more explicit given in terms of the energy’s variables and conjugate vari- [20,21]. Both are continuum mechanical models that, starting ables. (The conjugate variables are the energy derivatives. If from momentum conservation, focus on the total stress σij. w is a function of entropy s, density ρ and strain εij,theyare Because an explicit expression appears impossible, incre- Temperature T ≡ ∂w/∂s, chemical potential µ ≡ ∂w/∂ρ, mental relations are constructed, expressing ∂t σij in terms of and elastic stress πij ≡−∂w/∂εij). σij, velocity gradient ∇ j vi and mass density ρ. (Any such The hydrodynamic approach [28,29] has been success- expression, either for σij or ∂t σij, is referred to as a constitu- fully employed to account for many condensed systems, tive relation). Typically, neither elasto-plastic nor hypoplastic including liquid crystals [30–36], superfluid 3He [37–42], theory considers energy conservation. superconductors [43–45], macroscopic electro-magnetism The continuum mechanical formalism laid down by [46–49] and ferrofluids [50–58]. Transiently elastic media Truesdell and others [22–24] spells out a number of con- such as polymers are under active consideration at pres- straints, such as material objectivity, material symmetry, and ent [59–62]. Constructing a granular hydrodynamic theory, Clausius–Duhem (or entropy) inequality, which any phys- we believe, is both useful and possible: useful, because it ically sound constitutive relation must satisfy. The remain- should help to illuminate and order the complex macroscopic ing, still considerable discretion is deemed necessary to behavior of granular solid; possible, because total energy is account for the variation between different materials, say conserved in granular media, as it is in any other system. simple fluids and elastic solids. This is in contrast to the When comparing agitated sand to molecular gas, it is fre- hydrodynamic theory, a powerful approach to macroscopic quently emphasized that the kinetic energy, although con- field theories pioneered by Landau and Lifshitz [25,26] and served in the latter system, is not in the former, because the Khalatnikov [27], in the context of superfluid helium. By grains collide inelastically. This is undoubtedly true, but it considering energy and momentum conservation simulta- does not rule out the conservation of total energy, which neously, and combining both with thermodynamic consider- includes the entropy that accounts for the heat in the grains, ations, this approach cogently deduces the proper constitutive and in the air (or liquid) between them. To construct granu- relation—for a given energy expression. In other words, if the lar hydrodynamics, we need to start from some assumptions energy expression is known, the inclusion of energy conser- about the essence of granular physics, in order to specify vation adds so much constraints for the constitutive relation the energy w, its variables and the functional dependence. that it becomes unique. The modeling discretion is hence re- Our choice is given below, and argued for throughout this duced to a scalar energy expression, which, however, is quite manuscript. Aiming to be both general and specific, we first sufficient to account for material-specific differences, such specify a simple yet fairly realistic energy density, then derive as between simple fluids and elastic solids. Energy conserva- the structure of granular hydrodynamics that remains valid tion is also considered in continuum mechanics, see, e.g. [24], for more complex energies. The full theory is presented here, but the discussion does not go far enough to achieve a similar to prod the community to respond, and to serve as a reference degree of cogency. for our future works, in which we shall zoom into varying The total energy w depends, in addition to the relevant aspects and details of the theory, comparing them with exper- macroscopic variables such as the density ρ and the strain εij, iments. Only then will we be sure whether our assumptions always on the entropy density s: There are different though are appropriate, whether the presented set of partial differen- equivalent ways to understand s. The appropriate one here is tial equations is indeed the proper granular hydrodynamics. to take it as the summary variable for all implicit, microscopic Next a rough sketch of what we believe is the basic physics degrees of freedom. So the energy change associated with s, of granular behavior: granular motion may be divided into written as (∂w/∂s)ds ≡ T ds, is the increase of energy con- two parts, the macroscopic one arising from the large-scaled, tained in these degrees of freedom—what we usually refer smooth velocity of the medium, and the mesoscopic one from to as heat increase. When the macroscopic energy (such as the small-scaled, stochastic movements of the grains. The the elastic contribution) dissipates, it is being transferred into first is as usual accounted for by the hydrodynamic variable the microscopic degrees of freedom. The associated change of velocity, the second we shall account for by a scalar, the in entropy is such that the increase in heat is equal to the granular temperature Tg—although the analogy to molecu- loss of macroscopic energy, with the total energy w being lar motion is quite imperfect, the grains do not typically have conserved. velocities with a Gaussian distribution,